Singularities of Mean Curvature Flow with Bounded Total Curvature

We prove that every tangent flow at the first singular time of a smooth, closed, embedded mean curvature flow has smooth support, provided that the total curvature satisfies ${\sup }_{t\in [0,T)}{\int }_{M_t}\mathrm{\mid }A{\mathrm{\mid }}^{n}d{\mu }_t<\mathrm{\infty }$. This extends Ilmanen’s smoothness result for tangent flows to higher dimensions.